Omega appears and tells you that in an alternate reality, you believe that 2+2=3 with the same amount of credence, and asks whether this changes your own amount of credence that 2+2=4.
The answer is the same. You ask Omega what rules he’s playing by.
If he says “I’m visiting you in every reality. In each reality, I’m selecting a counterfactual where your answer is different” then you say “I have no new information, anthropic or otherwise, so I do not update.”
If he says “I’m visiting you in every reality. In each reality, I’m selecting a random alternate reality where you exist and telling you what that you believes” then you say “It’s equally likely that you randomly picked a reality where I am deceived and that I am in a reality where I am deceived. Therefore, I now give ‘2+2=4’ negligibly less than a .5 chance of being true.′
You are not asked to update your belief about the answer being “even” upon observing Omega (in any sense of “knowledge” of those discussed in the post). You knew that the other possibility existed all along, you don’t need Omega to see that. You are asked to decide what to do in the counterfactual.
Consider uncertainty about when Omega visits you part of the problem statement, but clearly if a tricky condition such as “it only visits you when your decision will make it worse for you” was assumed, it would be stated.
Suppose you believe that 2+2=4, with the caveat that you are aware that there is some negligible but non-zero probability that The Dark Lords of the Matrix have tricked you into believing that.
Omega appears and tells you that in an alternate reality, you believe that 2+2=3 with the same amount of credence, and asks whether this changes your own amount of credence that 2+2=4.
The answer is the same. You ask Omega what rules he’s playing by.
If he says “I’m visiting you in every reality. In each reality, I’m selecting a counterfactual where your answer is different” then you say “I have no new information, anthropic or otherwise, so I do not update.”
If he says “I’m visiting you in every reality. In each reality, I’m selecting a random alternate reality where you exist and telling you what that you believes” then you say “It’s equally likely that you randomly picked a reality where I am deceived and that I am in a reality where I am deceived. Therefore, I now give ‘2+2=4’ negligibly less than a .5 chance of being true.′
You are not asked to update your belief about the answer being “even” upon observing Omega (in any sense of “knowledge” of those discussed in the post). You knew that the other possibility existed all along, you don’t need Omega to see that. You are asked to decide what to do in the counterfactual.
Consider uncertainty about when Omega visits you part of the problem statement, but clearly if a tricky condition such as “it only visits you when your decision will make it worse for you” was assumed, it would be stated.